Question 1: Solve for x: x^2 + 6x + 5 = 0 using the completing the square method.

Answer: x = -1, -5

Question 2: Rewrite the quadratic equation 2x^2 + 8x - 10 = 0 in vertex form by completing the square.

Answer: 2(x + 2)^2 - 18 = 0

Question 3: Find the solutions of x^2 - 4x - 21 = 0 using the alternative completing the square method. Show all steps.

Answer: x = 7, -3

Question 4: Construct a quadratic equation with roots at x=2 and x=-3, then solve it by completing the square.

Answer: Equation: x^2 + x - 6 = 0; solutions: x=2, x=-3

Question 5: Verify the solution x=4 for the equation 3x^2 - 24x + 48 = 0 using the completing the square method.

Answer: Yes, x=4 is a root.

Question 6: Solve 5x^2 + 20x + 15 = 0 by completing the square. Show all working.

Answer: x = -1, -3

Question 7: A ball is projected such that its height h (in meters) after t seconds is given by h = -4.9t^2 + 3t + 2. Find the time when the ball hits the ground by solving the quadratic using completing the square.

Answer: t ≈ 0.43 seconds and t ≈ 0.85 seconds

Question 8: Identify and correct the mistake in solving x^2 + 4x + 1 = 0 by completing the square if the student incorrectly arrived at x = -2 ± √3.

Answer: Incorrect; the mistake was completing the square incorrectly. Correct form: (x + 2)^2 - 3 = 0, solutions: x = -2 ± √3

Question 9: Extend the method: Solve 3x^2 + 12x + 7 = 0 by completing the square, and explain each step.

Answer: 3(x + 2)^2 - 5 = 0; solutions: x = -2 ± √(5/3)

Question 10: Determine the roots of the quadratic by completing the square: x^2 + 10x + 25 = 0. Is this a perfect square?

Answer: x = -5 (repeated root), yes it is a perfect square.

Question 11: Challenge: Solve the quadratic equation 4x^2 + 4x + 1 = 0 using completing the square, and interpret the nature of the roots.

Answer: x = -0.5 (double root); roots are real and equal